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      numpy矩阵入门操作
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        <h2 id="numpy矩阵"><a href="#numpy矩阵" class="headerlink" title="numpy矩阵"></a>numpy矩阵</h2><p>矩阵是numpy.matrix类型的对象，该类继承自numpy.ndarray，任何针对多维数组的操作，对矩阵同样有效，但是作为子类矩阵又结合其自身的特点，做了必要的扩充，比如：乘法计算、求逆等。</p>
<h3 id="1-矩阵对象的创建"><a href="#1-矩阵对象的创建" class="headerlink" title="1. 矩阵对象的创建"></a>1. 矩阵对象的创建</h3><ol>
<li>通过ndarray创建matrix对象</li>
</ol>
<figure class="highlight python"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br></pre></td><td class="code"><pre><span class="line">numpy.matrix(</span><br><span class="line">    ary,		<span class="comment"># 任何可被解释为矩阵的二维容器</span></span><br><span class="line">  	copy=<span class="literal">True</span>	<span class="comment"># 是否复制数据(缺省值为True，即复制数据)</span></span><br><span class="line">)</span><br></pre></td></tr></table></figure>

<ol start="2">
<li><code>numpy.mat()</code><figure class="highlight python"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br></pre></td><td class="code"><pre><span class="line"><span class="comment"># 等价于：numpy.matrix(..., copy=False)</span></span><br><span class="line"><span class="comment"># 由该函数创建的矩阵对象与参数中的源容器一定共享数据，无法拥有独立的数据拷贝</span></span><br><span class="line">numpy.mat(任何可被解释为矩阵的二维容器)</span><br><span class="line"></span><br><span class="line"><span class="comment"># 该函数可以接受字符串形式的矩阵描述：</span></span><br><span class="line"><span class="comment"># 数据项通过空格分隔，数据行通过分号分隔。例如：'1 2 3; 4 5 6'</span></span><br><span class="line">numpy.mat(拼块规则)</span><br></pre></td></tr></table></figure>

</li>
</ol>
<p><strong>示例</strong></p>
<figure class="highlight python"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br></pre></td><td class="code"><pre><span class="line"><span class="comment"># 创建matrix操作</span></span><br><span class="line"><span class="keyword">import</span> numpy <span class="keyword">as</span> np</span><br><span class="line"></span><br><span class="line">arr = np.arange(<span class="number">1</span>, <span class="number">10</span>).reshape(<span class="number">3</span>, <span class="number">3</span>)</span><br><span class="line">print(arr)</span><br><span class="line"></span><br><span class="line"><span class="comment"># 第一种方式</span></span><br><span class="line">m = np.matrix(arr, copy=<span class="literal">True</span>)</span><br><span class="line">print(m, m.shape, type(m))</span><br><span class="line"></span><br><span class="line"><span class="comment"># 第二种方式:共享方式</span></span><br><span class="line">m2 = np.mat(arr)</span><br><span class="line"></span><br><span class="line"><span class="comment"># 第三种方式</span></span><br><span class="line">m3 = np.mat(<span class="string">"1 2 3;4 5 6.0"</span>)</span><br></pre></td></tr></table></figure>

<h3 id="2-矩阵的乘法运算"><a href="#2-矩阵的乘法运算" class="headerlink" title="2. 矩阵的乘法运算"></a>2. 矩阵的乘法运算</h3><figure class="highlight python"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br></pre></td><td class="code"><pre><span class="line"><span class="comment"># 矩阵乘法</span></span><br><span class="line"><span class="keyword">import</span> numpy <span class="keyword">as</span> np</span><br><span class="line"></span><br><span class="line">arr = np.array([[<span class="number">1</span>, <span class="number">1</span>, <span class="number">1</span>],</span><br><span class="line">                [<span class="number">2</span>, <span class="number">2</span>, <span class="number">2</span>],</span><br><span class="line">                [<span class="number">3</span>, <span class="number">3</span>, <span class="number">3</span>]])</span><br><span class="line"><span class="comment"># 数组相乘, 各对应位置元素相乘</span></span><br><span class="line">print(arr * arr)</span><br><span class="line"></span><br><span class="line"><span class="comment"># 矩阵相乘，第n行乘m列之和，作为结果的n,m个元素</span></span><br><span class="line"><span class="comment"># 矩阵相乘，第一个矩阵行数必须等于第二个矩阵列数</span></span><br><span class="line">m = np.mat(arr)</span><br><span class="line">print(m * m)</span><br></pre></td></tr></table></figure>

<h3 id="3-矩阵的逆矩阵"><a href="#3-矩阵的逆矩阵" class="headerlink" title="3. 矩阵的逆矩阵"></a>3. 矩阵的逆矩阵</h3><p>若两个矩阵A、B满足：AB = E （E为单位矩阵），则称B为A的逆矩阵。</p>
<p><strong>单位矩阵</strong></p>
<ul>
<li>在矩阵的乘法中，有一种矩阵起着特殊的作用，如同数的乘法中的1，这种矩阵被称为单位矩阵。</li>
<li>它是个方阵，从左上角到右下角的对角线（称为主对角线）上的元素均为1，除此以外全都为0，记为$I_n$或$E_n$ ，通常用I或E来表示。</li>
<li>根据单位矩阵的特点，任何矩阵与单位矩阵相乘都等于本身，而且单位矩阵因此独特性有广泛用途。<br>$$<br>E_3 =<br>\left[ \begin{array}{ccc}<br>1 &amp; 0 &amp; 0\<br>0 &amp; 1 &amp; 0\<br>0 &amp; 0 &amp; 1\<br>\end{array}<br>\right ]<br>$$</li>
</ul>
<a id="more"></a>

<p><strong>逆矩阵示例：</strong></p>
<figure class="highlight python"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br></pre></td><td class="code"><pre><span class="line">e = np.mat(<span class="string">"1 2 6; 3 5 7; 4 8 9"</span>)</span><br><span class="line">print(e.I)</span><br><span class="line">print(e * e.I)</span><br><span class="line"></span><br><span class="line"><span class="comment"># 非方阵的逆（称为广义逆矩阵）</span></span><br><span class="line">e = np.mat(<span class="string">"1 2 6; 3 5 7"</span>)</span><br><span class="line">print(e.I)</span><br><span class="line">print(e * e.I)</span><br></pre></td></tr></table></figure>

<p>注意：在计算过程中，可能出现<code>numpy.linalg.LinAlgError: Singular matrix</code>错误，说明该矩阵不可逆。</p>
<h3 id="4-ndarray提供的矩阵API"><a href="#4-ndarray提供的矩阵API" class="headerlink" title="4. ndarray提供的矩阵API"></a>4. ndarray提供的矩阵API</h3><p>ndarray提供了方法让多维数组替代矩阵的运算： </p>
<figure class="highlight python"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br></pre></td><td class="code"><pre><span class="line">a = np.array([</span><br><span class="line">    [<span class="number">1</span>, <span class="number">2</span>, <span class="number">6</span>],</span><br><span class="line">    [<span class="number">3</span>, <span class="number">5</span>, <span class="number">7</span>],</span><br><span class="line">    [<span class="number">4</span>, <span class="number">8</span>, <span class="number">9</span>]])</span><br><span class="line"><span class="comment"># 点乘法求ndarray的点乘结果，与矩阵的乘法运算结果相同</span></span><br><span class="line">k = a.dot(a)</span><br><span class="line">print(k)</span><br><span class="line"><span class="comment"># linalg模块中的inv方法可以求取a的逆矩阵</span></span><br><span class="line">l = np.linalg.inv(a)</span><br><span class="line">print(l)</span><br></pre></td></tr></table></figure>

<p><strong>执行结果：</strong></p>
<figure class="highlight python"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br></pre></td><td class="code"><pre><span class="line">[[ <span class="number">31</span>  <span class="number">60</span>  <span class="number">74</span>]</span><br><span class="line"> [ <span class="number">46</span>  <span class="number">87</span> <span class="number">116</span>]</span><br><span class="line"> [ <span class="number">64</span> <span class="number">120</span> <span class="number">161</span>]]</span><br><span class="line">[[<span class="number">-0.73333333</span>  <span class="number">2.</span>         <span class="number">-1.06666667</span>]</span><br><span class="line"> [ <span class="number">0.06666667</span> <span class="number">-1.</span>          <span class="number">0.73333333</span>]</span><br><span class="line"> [ <span class="number">0.26666667</span>  <span class="number">0.</span>         <span class="number">-0.06666667</span>]]</span><br></pre></td></tr></table></figure>

<h3 id="5-矩阵应用"><a href="#5-矩阵应用" class="headerlink" title="5. 矩阵应用"></a>5. 矩阵应用</h3><p><strong>案例：解线性方程组</strong></p>
<p>假设一帮孩子和家长出去旅游，去程坐的是bus，小孩票价为3元，家长票价为3.2元，共花了118.4；回程坐的是Train，小孩票价为3.5元，家长票价为3.6元，共花了135.2。分别求小孩和家长的人数。使用矩阵求解。表达成方程为：<br>$$<br>3x + 3.2y = 118.4\<br>3.5x + 3.6y = 135.2<br>$$<br>表示成矩阵相乘：<br>$$<br>\left[ \begin{array}{ccc}<br>    3 &amp; 3.2 \<br>    3.5 &amp; 3.6 \<br>\end{array} \right]<br>\times<br>\left[ \begin{array}{ccc}<br>    x \<br>    y \<br>\end{array} \right]<br>=<br>\left[ \begin{array}{ccc}<br>    118.4 \<br>    135.2 \<br>\end{array} \right]<br>$$</p>
<figure class="highlight python"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br></pre></td><td class="code"><pre><span class="line"><span class="keyword">import</span> numpy <span class="keyword">as</span> np</span><br><span class="line"></span><br><span class="line"><span class="comment"># 解方程</span></span><br><span class="line">prices = np.mat(<span class="string">'3 3.2; 3.5 3.6'</span>)</span><br><span class="line">totals = np.mat(<span class="string">'118.4; 135.2'</span>)</span><br><span class="line"></span><br><span class="line">x = np.linalg.lstsq(prices, totals)[<span class="number">0</span>]  <span class="comment"># 求最小二乘解</span></span><br><span class="line">print(x)</span><br><span class="line"></span><br><span class="line">x = np.linalg.solve(prices, totals)  <span class="comment"># 求解线性方程的解</span></span><br><span class="line">print(x)</span><br><span class="line"></span><br><span class="line">x = prices.I * totals  <span class="comment"># 利用矩阵的逆进行求解</span></span><br><span class="line">print(x)</span><br></pre></td></tr></table></figure>

<p><strong>案例：斐波那契数列</strong></p>
<p>1    1     2     3    5    8    13    21    34 …</p>
<figure class="highlight python"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br></pre></td><td class="code"><pre><span class="line">X    |  <span class="number">1</span>   <span class="number">1</span>  |  <span class="number">1</span>   <span class="number">1</span>  |  <span class="number">1</span>   <span class="number">1</span></span><br><span class="line">     |  <span class="number">1</span>   <span class="number">0</span>  |  <span class="number">1</span>   <span class="number">0</span>  |  <span class="number">1</span>   <span class="number">0</span></span><br><span class="line">    --------------------------------</span><br><span class="line"><span class="number">1</span>  <span class="number">1</span> |  <span class="number">2</span>   <span class="number">1</span>  |  <span class="number">3</span>   <span class="number">2</span>  |  <span class="number">5</span>   <span class="number">3</span></span><br><span class="line"><span class="number">1</span>  <span class="number">0</span> |  <span class="number">1</span>   <span class="number">1</span>  |  <span class="number">2</span>   <span class="number">1</span>  |  <span class="number">3</span>   <span class="number">2</span></span><br><span class="line"> F^<span class="number">1</span>     F^<span class="number">2</span>       F^<span class="number">3</span> 	     F^<span class="number">4</span>  ...  f^n</span><br></pre></td></tr></table></figure>

<figure class="highlight python"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br></pre></td><td class="code"><pre><span class="line"><span class="keyword">import</span> numpy <span class="keyword">as</span> np</span><br><span class="line">n = <span class="number">35</span></span><br><span class="line"><span class="comment"># 使用递归实现斐波那契数列</span></span><br><span class="line"><span class="function"><span class="keyword">def</span> <span class="title">fibo</span><span class="params">(n)</span>:</span></span><br><span class="line">    <span class="keyword">return</span> <span class="number">1</span> <span class="keyword">if</span> n &lt; <span class="number">3</span> <span class="keyword">else</span> fibo(n - <span class="number">1</span>) + fibo(n - <span class="number">2</span>)</span><br><span class="line">print(fibo(n))</span><br><span class="line"></span><br><span class="line"><span class="comment"># 使用矩阵实现斐波那契数列</span></span><br><span class="line">print(int((np.mat(<span class="string">'1. 1.; 1. 0.'</span>) ** (n - <span class="number">1</span>))[<span class="number">0</span>, <span class="number">0</span>]))</span><br></pre></td></tr></table></figure>


      
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